LINEAR EQUATIONS IN TWO VARIABLES FOR CBSE-NCERT

Topic Covered

Equations of Lines Parallel to the x-axis and y-axis

You have studied how to write the coordinates of a given point in the Cartesian plane.

Do you know where the points `(2, 0), (–3, 0), (4, 0)` and `(n, 0)`, for any real number n, lie in the Cartesian plane? Yes, they all lie on the x-axis. But do you know why? Because on the x-axis, the y-coordinate of each point is `0`.

In fact, every point on the x-axis is of the form `(x, 0)`. Can you now guess the equation of the x-axis? It is given by `y = 0`. Note that `y = 0` can be expressed as `0.x + 1.y = 0`. Similarly, observe that the equation of the y-axis is given by `x = 0`.

Now, consider the equation `x – 2 = 0`. If this is treated as an equation in one variable x only, then it has the unique solution `x = 2`, which is a point on the number line. However, when treated as an equation in two variables, it can be expressed as

`x + 0.y – 2 = 0`. This has infinitely many solutions. In fact, they are all of the form `(2, r)`, where `r` is any real number. Also, you can check that every point of the form (2, r) is a solution of this equation.

So as, an equation in two variables, `x – 2 = 0` is represented by the line AB in the graph in Fig. 4.8.

You have studied how to write the coordinates of a given point in the Cartesian plane.

Do you know where the points `(2, 0), (–3, 0), (4, 0)` and `(n, 0)`, for any real number n, lie in the Cartesian plane? Yes, they all lie on the x-axis. But do you know why? Because on the x-axis, the y-coordinate of each point is `0`.

In fact, every point on the x-axis is of the form `(x, 0)`. Can you now guess the equation of the x-axis? It is given by `y = 0`. Note that `y = 0` can be expressed as `0.x + 1.y = 0`. Similarly, observe that the equation of the y-axis is given by `x = 0`.

Now, consider the equation `x – 2 = 0`. If this is treated as an equation in one variable x only, then it has the unique solution `x = 2`, which is a point on the number line. However, when treated as an equation in two variables, it can be expressed as

`x + 0.y – 2 = 0`. This has infinitely many solutions. In fact, they are all of the form `(2, r)`, where `r` is any real number. Also, you can check that every point of the form (2, r) is a solution of this equation.

So as, an equation in two variables, `x – 2 = 0` is represented by the line AB in the graph in Fig. 4.8.

(i) The representation of the solution on the number line is shown in Fig. 4.9, where
`x = – 4` is treated as an equation in one variable.

(ii) We know that `x = – 4` can be written as
`x + 0.y = – 4`

which is a linear equation in the variables x and y. This is represented by a line. Now all the values of y are permissible because 0.y is always 0.

However, x must satisfy the equation x = – 4. Hence, two solutions of the given equation are x = – 4, y = 0 and x = – 4, y = 2.
Note that the graph AB is a line parallel to the y-axis and at a distance of 4 units to the left of it (see Fig. 4.10).

Similarly, you can obtain a line parallel to the x-axis corresponding to equations of the type